# Linear Algebra

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Linear Algebra

## Definiteness

A square matrix is positive definite if for all vectors , .

If the inequality is weak () then the matrix is positive semi-definite.

### Properties

- If the determinant of every upper-left submatrix is positive then the matrix is positive-definite.
- If is PSD then is PSD.
- The sum of PSD matrices is PSD.

### Examples

Examples of PSD matrices:

- The identity matrix is PSD
- is PSD for any vector